Remke Kloosterman
In this paper we classify all configurations of singular fibers of elliptic fibrations on the double cover of P^2 ramified along six lines in general position.
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In this paper we give the Weierstrass equations and the generators of Mordell-Weil groups for Jacobian fibrations on the singular K3 surface of discriminant 3.
In this paper we give the Weierstrass equations for Jacobian fibrations on the K3 surface that is the minimal resolution of the double covering of projective plane ramified along generic six lines.
For a general cubic fourfold, it was observed by Donagi and Markman that the relative intermediate Jacobian fibration associated to the family of its hyperplane sections carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification of the intermediate Jacobian fibration giving a ten-dimensional compact hyper-Kähler manifold, which we then show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture by Markushevich.
The Intermediate Jacobian fibration of a cubic fourfold containing a plane and fibrations in Prym varieties
本文描述了含平面的三次四fold的中间雅可比纤维化，将其视为K3曲面上挠层模空间的Lagrangian子纤维化，并提出了Prym簇中Lagrangian纤维化的一般构造。
The starting point of this note is our recent paper with Laza and Saccà on the construction of deformations of O'Grady's $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds. The note provides a complement to that paper consisting in the analogous construction in the twisted case, leading to isogenous but presumably not isomorphic or birational hyper-Kähler manifolds.
Jacobian elliptic fibrations on K3s with a non-symplectic automorphism of order 3
本文依据3阶非辛自同构对纤维的作用，结合Kneser–Nishiyama方法分类相应K3曲面上的Jacobian椭圆纤维化，并给出可约纤维类型与σ不变Weierstrass方程。
We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell-Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.
Let Y->P^n be a flat family of reduced Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that X is a Beauville-Mukai integrable system if n=3, 4, or 5, and the curves are irreducible and non-hyperelliptic. We also prove that X is a Beauville-Mukai system if n=3, d is odd, and the curves are canonically positive 2-connected hyperelliptic curves.
We prove that if $X$ and $S$ are smooth varieties and $f\colon X\to S$ is an elliptic fibration with singular fibers curves of types I$_N$ with $N\geq 1$, II, III and IV, then the relative Jacobian $\hat{f}\colon \bar{M}_{X/S}\to S$ of $f$, defined as the relative moduli space of semistable pure dimension one sheaves of rank 1 and degree 0 on the fibers of $f$, is an elliptic fibration such that all its fibers are irreducible. This extends known results when fibers are integral or of type I$_2$.
Sections of Jacobian fibrations over lines
本文研究光滑曲面线性系中直线对应的通用雅可比截面的上同调类，针对K3曲面的通用紧化雅可比，修正了排印错误与猜想7，探讨结果与文献步骤的协调问题。
Let Y->P^n be a flat family of integral Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that the degree of the discriminant locus Delta in P^n is at least 4n+2, and we prove that X is a Beauville-Mukai integrable system if the degree of Delta is greater than 4n+20.
Compactified Jacobians of Extended ADE Curves and Lagrangian Fibrations
本文提出扩展 ADE 曲线概念，描述其紧化雅可比簇分支与相交图的关系，并给出 Beauville-Mukai 型拉格朗日纤维化一般奇异纤维的显式描述。
Jacobian elliptic fibrations on the generalized Inose quartic of Picard rank sixteen
本文研究 Picard 格包含 $H \oplus E_7(-1) \oplus E_7(-1)$ 的复代数 K3 曲面族，证明其一般成员恰有四个互不等价的 Jacobian 椭圆纤维化，并构造了显式线束。
On Néron-Severi lattices of Jacobian elliptic K3 surfaces
本文分类了有限自同构群 K3 曲面及无限自同构群且 Néron-Severi 格为 2-初等情形下具有有限 Mordell-Weil 群的 Jacobian 椭圆纤维化，并计算了相应格论重数。
The defect of a cubic threefold and applications to intermediate Jacobian fibrations
本文研究具有孤立奇点的三次三流形的缺陷不变量，利用过奇点的直线曲线拓扑数据计算缺陷，刻画混合霍奇结构，并证明非Q-阶乘性等价于包含平面或三次直纹面，进而联系到三次四流形的中间雅可比纤维化。
Twists of intermediate Jacobian fibrations
本文通过引入解析相对雅可比层，研究OG10超凯勒拉格朗日量的截面、Tate–Shafarevich扭曲与周期，并用解析Brauer群计算扭曲群，证明三次四维流形的本原Hodge格与OG10流形第二上同调的特殊子格等距。
Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
该研究证明了任意光滑三次四fold的中等雅可比纤维化具有超凯勒紧化，并计算了其双有理几何中的锥，同时确定了Mordell-Weil群与代数上同调的同构关系。
We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, $R_{5,5}$. Such surfaces have a natural elliptic fibration induced by the fibration on $R_{5,5}$. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on $R_{5,5}$. This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in de…
We construct examples of elliptic fibrations of orbifold general type (in the sense of Campana) which have no etale covers dominating a variety of general type.
We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves…
We study K3 surfaces over a number field $k$ which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus 1 fibration.
In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron--Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibr…
We consider t-structures that naturally arise on elliptic fibrations. By filtering the category of coherent sheaves on an elliptic fibration using the torsion pairs corresponding to these t-structures, we prove results describing equivalences of t-structures under Fourier-Mukai transforms.
We pose the problem to determine explicit defining equations of various elliptic fibrations on a given $K3$ surface, and study the case of the Kummer surfaces of the product of two elliptic curves.
A fibration is said to be isotrivial if all of its smooth fibres are isomorphic to a single fixed variety. We classify the elliptic K3 surfaces that are isotrivial, and use them to construct Lagrangian fibrations that are isotrivial. We then modify the construction to produce new examples of holomorphic symplectic orbifolds, that also admit isotrivial Lagrangian fibrations.
We prove rationality results for moduli spaces of elliptic K3 surfaces and elliptic rational surfaces with fixed monodromy groups.
This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptiona…
Elliptic and K3 fibrations on double octic Calabi-Yau threefolds
研究由八次排列的奇异线和重数≥4的点确定的双八次Calabi-Yau三重折叠的椭圆与K3纤维化，得出每个双八次均存在多类纤维化，并据此构建了两个双八次家族元素间的双有理映射。
In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of $\mathbb{P}^2$ branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen--Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen--Sarti involution induces a 2-isogeny between two K3 surfaces, which is described in this paper.
We describe all the elliptic models with section on the Shioda supersingular K3 surface X of Artin invariant 1 over an algebraically closed field of characteristic 3. In particular, we compute elliptic parameters and Weierstrass equations for the fifty two different fibrations, and analyze the reducible fibers and Mordell-Weil lattices.
In this work we describe a construction of semistable fibrations over an elliptic curve with one unique singular fibre and we give effective examples using monodromy of curves.
We study the Gromov-Witten and Donaldson-Thomas correspondence conjectured in [MNOP1, MNOP2], for trivial elliptic fibrations. In particular, we verify the Gromov-Witten and Donaldson-Thomas correspondence for primary fields when the threefold is $E \times S$ where $E$ is a smooth elliptic curve and $S$ is a smooth surface with numerically trivial canonical class.
In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes ${\cal H}(1,X(d),n)$ of morphisms of degree $n$ from elliptic curves to the modular curve $X(d)$, $d\geq 3$. Ultimately, we show that the moduli spaces in…
This is mainly a review of my results related to the title. We discuss, how many elliptic fibrations and elliptic fibrations with infinite automorphism group (or the Mordell-Weil group) an algebraic K3 surface over an algebraically closed field can have. This was the subject of my talk at Oberwolfach Workshop "Higher dimensional elliptic fibrations" in October 2010.
We define a monodromy homomorphism for irreducible families of regular elliptic fibrations which takes values in the mapping class group of a punctured sphere. We compute the monodromy for elliptic fibrations only which contain no singular fibres of types other than I_1 and I*_0. There are maximal groups, which can be the monodromy groups of algebraic, resp. differentiable families of elliptic surfaces, and we give an algebraic criterion for the equality of both groups which we can check to apply in case the number of I_1 fibres is at most 6.
Elliptic fibrations and involutions on K3 surfaces
本文综述了带有非辛对合的 K3 曲面上椭圆纤维化的分类，在更一般的对合框架下重新审视了具有 2-元初 Néron--Severi 格的 K3 曲面的完整分类，并给出了有理椭圆曲面二次覆盖的 K3 曲面上极值椭圆纤维化的完整分类。
We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. By generalizing \cite{EOY14}, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree $22$.
We classify complex K3 surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers. These surfaces are characterized by the fact that they admit a unique elliptic fibration with infinite automorphism group. We furnish an explicit list of 32 Néron-Severi lattices corresponding to such surfaces. Incidentally, we are able to decide which of these 32 classes of surfaces admit a unique genus 1 fibration. Finally, we prove that all K3 surfaces with Picard rank >=19 and infinite automorphism group have positive entropy.
We consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah's and Tu's results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree.
Counting elliptic fibrations on K3 surfaces
本文研究任意复射影 K3 曲面上模自同构的 Jacobian 椭圆纤维化计数问题，给出求解方法并以显式示例加以说明。
This paper is concerned with the construction of extremal elliptic K3 surfaces. It gives a complete treatment of those fibrations which can be derived from rational elliptic surfaces by easy manipulations of their Weierstrass equations. In particular, this approach enables us to find explicit equations for 38 semi-stable extremal elliptic K3 fibrations, 32 of which are indeed defined over the rationals. They are realized as pull-back of non-semi-stable extremal rational elliptic surfaces via base change. This is related to work of J. Top and N. Yui which exhibited the same procedure for the se…
Given a non-singular variety with a K3 fibration f : X --> S we construct dual fibrations Y --> S by replacing each fibre X_s of f by a two-dimensional moduli space of stable sheaves on X_s. In certain cases we prove that the resulting scheme Y is a non-singular variety and construct an equivalence of derived categories of coherent sheaves Φ: D(Y) --> D(X). Our methods also apply to elliptic and abelian surface fibrations. As an application we show how the equivalences Φidentify certain moduli spaces of stable bundles on elliptic threefolds with Hilbert schemes of curves.
The theorem referred to in the title is a technical result that is needed for the classification of elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces in weighted projective space. We present a complete proof of the Curve Exclusion Theorem, which first appeared in the author's unpublished PhD thesis [Ry02] and has since been relied upon in solutions to many cases of the fibration classification problem. We give examples of these solutions and discuss them briefly.
We investigate Calabi--Yau three folds which are small resolutions of fiber products of elliptic surfaces with section admitting reduced fibers. We start by the classification of all fibers that can appear on such varieties. Then, we find formulas to compute the Hodge numbers of obtained three folds in terms of the types of singular fibers of the elliptic surfaces. Next we study Kummer fibrations associated to these fiber products.
A complete classification is presented of elliptic and K3 fibrations birational to certain mildly singular complex Fano 3-folds. Detailed proofs are given for one example case, namely that of a general hypersurface X of degree 30 in weighted P^4 with weights 1,4,5,6,15; but our methods apply more generally. For constructing birational maps from X to elliptic and K3 fibrations we use Kawamata blowups and Mori theory to compute anticanonical rings; to exclude other possible fibrations we make a close examination of the strictly canonical singularities of (X,(1/n)H), where H is the linear system …
Bounds for the order of automorphism groups of cyclic covering fibrations of an elliptic surface
本文针对椭圆曲面的有限循环覆盖纤维化，利用纤维亏格、基曲线亏格、覆盖次数和相对典范除子平方，给出了自同构群有限子群阶的上界估计。
Singular fibers of elliptic fibrations on normal $K3$ surfaces
研究正规K3曲面上椭圆纤维化的奇异纤维，通过极小分辨率，依据\((-2)\) - 曲线的ADE构型收缩来描述，确定了相关构型及收缩后纤维，从而得到奇异纤维的描述。
We systematically develop Bridgeland's and Bridgeland-Maciocia's techniques for studying elliptic fibrations, and identify criteria that ensure 2-term complexes are mapped to torsion-free sheaves under a Fourier-Mukai transform. As an application, we construct an open immersion from a moduli of stable complexes to a moduli of Gieseker stable sheaves on elliptic threefolds. As another application, we give various 1-1 correspondences between fiberwise semistable torsion-free sheaves and codimension-1 sheaves on Weierstrass surfaces.
We study the quasimap invariants of elliptic and K3 fibrations. Oberdieck and Pixton conjectured that the Gromov-Witten potentials of elliptic fibrations are quasi-modular forms. Analogously, we propose similar conjecture for the quasimap potentials of elliptic fibrations. We also conjecture some finite generation properties of quasimap potentials of K3 fibrations. Via wall-crossing conjecture, this will imply some quasi-modularity of the Gromov-Witten potentials of K3 fibrations. We provide some evidences for our conjectures through several examples. The method here can be further generalized…
We consider elliptic fibrations with arbitrary base dimensions, and generalise previous work by the second author. In particular, we check universal closedness for the moduli of semistable objects with respect to a polynomial stability that reduces to PT-stability on threefolds. We also show openness of this polynomial stability. On the other hand, we write down criteria under which certain 2-term polynomial semistable complexes are mapped to torsion-free semistable sheaves under a Fourier-Mukai transform. As an application, we construct an open immersion from a moduli of complexes to a moduli…
Let $V_1$ be the Fano threefold given as a hypersurface of degree 6 in $P(1,1,1,2,3)$ (over a number field $K$). Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.
We present the topological classification of real parts of real regular elliptic surfaces with a real section.
Elliptic Fibrations on Vinberg's Most Algebraic K3 Surface
本文介绍如何利用计算机代数系统 OSCAR 寻找给定曲面上的所有椭圆纤维化并计算其 Weierstrass 模型，并以 Vinberg 最代数 K3 曲面为例进行了演示。
We characterize the subscheme of the moduli space of torsion-free sheaves on an elliptic surface which are stable of relative degree zeero (with respect to a polarization of type aH+bf, H being the section and f the elliptic fibre) which is isomorphic, via the relative Fourier-Mukai transform, with the relative compactified Simpson Jacobian of the family of those curves D in the surface which are flat over the base of the elliptic fibration. This generalizes and completes earlier constructions due to Friedman, Morgan and Witten. We also study the relative moduli scheme of sheaves whose restric…
We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_n^l$, which is the quotient of the product of a certain curve $C_n$ by itself $l$ times by a group $G\simeq\left(\mathbb{Z}/n\mathbb{Z}\right)^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved: the Kod…
We find an asymptotic description of the period locus of simply connected Jacobian elliptic surfaces and of the period locus of hyperelliptic curves. The two descriptions are essentially the same, and are given by the alkanes of organic chemistry.
In recent years there has been an interest in constructing examples of closed Riemann surfaces whose jacobian varieties are isogenous to a product of many elliptic factors and some other jacobian varieties. The first ones, provided by Ekedahl and Serre, are examples for which the isogenous decomposition has all factors being elliptic curves. It is well known that given two elliptic curves $E_{1}$ and $E_{2}$, there is a closed Riemann surface $X$ of genus two, with equations in terms of the elliptic curves, and whose jacobian variety $JX$ is isogenous to $E_{1} \times E_{2}$. In this paper, gi…
Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice $U^3\oplus E_8(-1)^2$ depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.
We prove a generic Torelli theorem for Jacobian elliptic surfaces, provided that the geometric genus is large compared to the irregularity. The result is effective to the extent that defining equations for the base curve are recovered from the period data.
527 elliptic fibrations on Enriques surfaces
本文证明每个 Enriques 曲面在适当多重性计数下都恰好有 527 个椭圆纤维化同构类，并通过构造模空间研究遗忘映射的分歧行为，将其与 Morrison-Kawamata 锥猜想中的双曲体积联系起来。
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